The current document is directed to methods and systems that sample log/event messages for downstream processing by log/event-message systems incorporated within distributed computer facilities. The data-collection, data-storage, and data-querying functionalities of log/event-message systems provide a basis for distributed log-analytics systems which, in turn, provide a basis for automated and semi-automated system-administration-and-management systems. By sampling log/event-messages, rather than processing and storing every log/event-message generated within a distributed computer system, a log/event-message system significantly decreases data-storage-capacity, computational-bandwidth, and networking-bandwidth overheads involved in processing and retaining large numbers of log/event messages that do not provide sufficient useful information to justify these costs. Increase in efficiencies of log/event-message systems obtained by sampling translate directly into increases in bandwidths of distributed computer systems, in general, and to increases in time periods during which useful log/event messages can be stored.